If the co-ordinate axes are rotated though though an π6 about the origin then the transformed of equation 3x2−4xy+3y2=0 is

X=xcos⁡θ−ysin⁡θ=x32−y2=3x-y2⇒Y=xsin⁡θ+ycos⁡θ=12x+32y=x+3y2Now transformed equation is sub X,Y in given equation we get 3y2-xy=0

If the co-ordinate axes are rotated though though an π6 about the origin then the transformed of equation 3x2−4xy+3y2=0 is

X=xcos⁡θ−ysin⁡θ=x32−y2=3x-y2⇒Y=xsin⁡θ+ycos⁡θ=12x+32y=x+3y2Now transformed equation is sub X,Y in given equation we get 3y2-xy=0

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If the co-ordinate axes are rotated though though an $\frac{π}{6}$ about the origin then the transformed of equation $\sqrt{3} x^{2} - 4 x y + \sqrt{3} y^{2} = 0$ is

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$\sqrt{3} y^{2} - 2 x y = 0$

$x^{2} - y^{2} = 0$

$\sqrt{3} y^{2} - x y = 0$

$\sqrt{3} y^{2} + x y = 0$

answer is C.

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Detailed Solution

$\begin{array}{r} X = x \cos θ - y \sin θ = x \frac{\sqrt{3}}{2} - \frac{y}{2} = \frac{\sqrt{3} x - y}{2} \\ \Rightarrow Y = x \sin θ + y \cos θ = \frac{1}{2} x + \frac{\sqrt{3}}{2} y = \frac{x + \sqrt{3} y}{2} \\ N o w t r a n s f o r m e d e q u a t i o n i s s u b X, Y i n g i v e n e q u a t i o n \\ w e g e t \sqrt{3} y^{2} - x y = 0 \end{array}$